arX

iv:1

011.

1642

v2 [

mat

h.C

A]

13

Jan

2012

Contemporary Mathematics 563 (2012), 1–31

Spectral/quadrature duality:Picard–Vessiot theory and finite-gap potentials

Yurii V. Brezhnev

Abstract. In the framework of differential Galois theory we treat the classi-cal spectral problem Ψ′′ −u(x)Ψ = λΨ and its finite-gap potentials as exactlysolvable in quadratures by Picard–Vessiot without involving special functions;the ideology goes back to the 1919 works by J. Drach. We show that duality be-tween spectral and quadrature approaches is realized through the Weierstrasspermutation theorem for a logarithmic Abelian integral. From this standpointwe inspect known facts and obtain new ones: an important formula for theΨ-function and Θ-function extensions of Picard–Vessiot fields. In particular,extensions by Jacobi’s θ-functions lead to the (quadrature) algebraically inte-grable equations for the θ-functions themselves.

Contents

1. Introduction 22. Background 53. Integrability of equation (1) by Picard–Vessiot 84. Spectral/quadrature duality. An integration procedure 125. The Θ-functons 166. Integration as a linearly exponential Θ-extension 197. Integrability and differential closedness 208. Definition of θ through Liouvillian extension 249. Non-finite-gap integrable counterexamples 2710. Concluding remarks 29References 30

Key words and phrases. Schrodinger equation, finite-gap potentials, Picard–Vessiot theory,quadratures, Liouvillian extensions, Abelian integrals, theta-functions.

Research supported by the Federal Targeted Program under contract 02.740.11.0238 andpartially by Royal Society/NATO and RFBR grant (00–01–00782).

c©0000 (copyright holder)

1

2 YU. BREZHNEV

1. Introduction

Initially the method of finite-gap integration was developed in works [48, 29,30, 39, 44, 22, 23] as a periodic generalization of the celebrated inverse scatteringtransform method (the soliton theory [46]). In the very first papers on this topic[48, 30, 41, 44, 22] it has become clear that analogs and generalizations of thesoliton potentials are to be smooth and real functions u(x) in the spectral problemdefined by the Schrodinger equation

(1) Ψxx − u(x)Ψ = λΨ ,

if the continuous spectrum of the problem consists of finitely many ‘forbidden gaps’1

(lacunae, bands, zones, intervals [44, 39]). This explains the widely used termi-nology ‘finite-gap’, abbreviated further as FG. Ensuing development of the the-ory went far beyond equation (1) and took the algebro-geometric characterization[25, 37, 38, 24]. Riemann surfaces and their theta-functions have become themain subject of study [41, 24, 25]. Appearance of these nontrivial objects is dic-tated by the very nontrivial dependence of solution to Eq. (1) upon parameter λand the search for this dependence is a starting and prime subject of the spectral[44, 29, 30] and algebro-geometric (Θ-function) [37, 24] approaches. These the-ories are referred frequently to as the Θ-function integration. Presently, one cansay that the intense study of equation (1) over the last decades led to the factthat its FG-theory has been developed almost exhaustively. In this connection, itis, perhaps, not without interest to consider one more view on integration of theproblem (1).

1.1. Motivation. In the early 1980s some authors revealed the two old pa-pers by J. Drach [20, 21] wherein equation (1) was integrated ‘directly’ and mainresults of the theory were presented in extremely condensed form. Although theseworks had subsequently received some mention in the literature [26, 17, 46] witha special emphasis to the FG-theory ([8, pp. 84–85]; written by Matveev), somesurprising facet is the fact that more detailed exposition of Drach’s ideology hasnot been presented in the modern literature hitherto. The need for such expositionis apparent when taken into account that the works2 [20, 21] themselves containno any explanations or proofs. In this connection, it is pertinent to make up forthis gap and sketch an appropriate theory.

The original approach by Drach is to integrate (1) as an ODE. Indeed, equation(1) is primarily a differential equation in variable x even though we consider itin the algebro-geometric [37, 7] or spectral context [30, 44] in which the Ψ isviewed, primarily, as function of λ. Anyway, in so far as the Ψ(x;λ) is a functionof two variables, complete theory must explain this duality and therefore provideconversion between ‘x-’ and ‘λ-formulae’. On the other hand, integration of linearODEs is the subject matter of the old and well developed differential extension ofthe algebraic Galois theory which is variously known as the Picard–Vessiot theoryand sometimes as the Lie–Kolchin theory. The main references in this topic aremonographs [50, 31, 9] and classical works [51, 32]. Strange though it may seem,the explicit discussion of a direct linkage between this theory and the modern

1If parameter λ belongs to such a lacunae (it is a line segment on the real axe λ), then theΨ-function growths unboundedly as a function of x.

2After these works Drach had not longer returned to integration of linear ODEs.

SPECTRAL/QUADRATURE DUALITY OF FINITE-GAP POTENTIALS 3

aspects of integrable models associated to Eq. (1) appeared comparatively recently[45, 2].

Correlation between the Picard–Vessiot theory and Θ-function methods bringsup the following question: what is the relation between these two techniques uponapplying them to the linear spectral problems, say, (1)? The formal answer (com-monly accepted (?)) might be the following: the Θ-series (see next section fordefinition) is a special function solving (1). Indeed, an integration theory by Picard–Vessiot begins with the precise definition of a used function class [50, 32]. By thisis meant that, in the strict sense, without definition of the integrability domain, anyscheme based on use of the formally postulated Θ-series should indeed be consid-ered as integration in terms of special functions. We shall show that this is not thecase. For example, the Θ-function integration of Hamiltonian finite-dimensionalnonlinear dynamical systems q = V (q) is know to be a manifestation of their al-gebraically invariant Liouvillian integrability. On the other hand, integrability ofsuch systems is very well known to be related [19] to their representability throughcertain linear equations (Lax pairs):

L(q)Ψ = λΨ , Ψ = A(q)Ψ

⇒ L =

[A,L

]⇔ q = V (q).

The ‘nonlinear Hamiltonian’ integrability is certain to entail the ‘linear Picard–Vessiot’ one, if only because the logarithmic derivative Ψ/Ψ is a rational function ofdynamical variables. It should be noted here that the development of this ideologycan serve as a basis for independent concept of integration of spectral problemsat all. However, we do not touch here on such Hamiltonian systems and algebro-geometric (FG) integration of partial differential equations (PDEs). We focus onlyon a linear problem as such, so our main intention with this work is to show thatthe scheme of FG-integration of the linear spectral equation (1) should be separatedinto the two parts:

(1) The invariant property of equation (1) to be integrable, i. e., Lie–Kolchin’ssolvability of corresponding differential Galois group [32, 50].

(2) Representation of differential fields and solutions in terms of those func-tions which of inevitably appear in the theory. These are the Θ-series. Insome particular cases the series themselves satisfy the algebraically inte-grable Hamiltonian ODEs.

By the invariance, here and in the subsequent discussion, we shall informallymean an independence of representations by theta-functions. Notwithstanding thefact that representation of solutions requires introducing the highly nontrivial tran-scendental Θ-objects, the integrability mechanism itself is very simple. It coincidesin effect with an elementary solvability in closed form3 and thereby trivializes un-derstanding of the major portion of the FG-theory. In other words, our mainpurpose is to bring the Picard–Vessiot aspects—fields, their extensions, differentialGalois group, quadrature solvability, etc—into the foreground and, subsequently, toget representations for them in the FG-terms—spectral curves, variables γk, Θ’s,etc. The latter objects, to the best of our knowledge, have not received mention

3J. Kovacic, in his famous work [35] on p. 4, notes: ‘by a “closed-form” solution we mean,roughly, one that can be written down by a first-year calculus student’. As we shall see, this‘definition’ is completely compatible with the transcendental Θ-function characterization of theFG-integration of Eq. (1). Rephrasing, there is a closed form solution (Theorem 4.2) that can beverified by a direct substitution into (1) followed by use of the first-year student calculus: algebraand differentiation.

4 YU. BREZHNEV

in the contemporary works on the Picard–Vessiot integration of linear ODEs. See,for example, work [49, first sentence in §3(b)], monographs [50, 45, 9], and quitevoluminous references therein. Partially, some fragments of the theory, in a contextof the elliptic Lame potentials u = A℘(x), can be found in book [45] and work[10]. We note also that FG-potentials are the Abelian functions [41] and Abelianextensions of differential fields were already briefly considered by Kolchin himself[33]. There is no escape from the mentioning nice applications of the Picard–Vessiottheory of Eq. (1) to the supersymmetric quantum mechanics. They appeared com-paratively recently and this theme is the subject matter of recent works [1, 2].

1.2. Outline of the work. Section 2 contains the background material: asketch of the classical Burchnall–Chaundy theory of commuting operators and itsmodern formulation in the language of theta-functions.

In Sect. 3 we briefly recall the needed facts from the Picard–Vessiot theory,introduce the base differential fields (Novikov’s fields), and motivate their hyperel-liptic extensions. Then we describe a structure of the differential Galois group forFG-potentials.

Section 4 is devoted to the quadrature (Drach) characterization of the FG-integrability and an explanation as to how the known Weierstrass theorem on anAbelian logarithmic integral performs the transition and difference between quad-rature and spectral mode of getting formulae for the Ψ-function.

In Sect. 5 we first recall that the theta-function formulae can be derived fromquadratures ones [13] and then show the necessity to represent the previous baseobjects in terms of theta-functions and, in particular, to introduce an importantobject—the 1-dimensional section of the theta-function argument with a free param-eter. Owing to some differential properties of theta-functions the theory acquiresvery effective form in those cases when jacobians are reducible to a product ofelliptic curves.

In Sect. 6 we completely pass to the theta-function representations and givean appropriate formulation to the FG-Picard–Vessiot integrability of Eq. (1). Thisprovides a nice analogy with solubilities in the simplest integrability domains likefield C.

Section 7 explains how the theta-function reformulation of Picard–Vessiot in-tegrability transforms into the differential closedness of the theta-functions them-selves. This also gives a new treatment to the spectral parameter and a relationshipof this treatment to the closedness and linearity of some of defining equations. Weexpound results at greater length for the cases when multi-dimensional Θ reducesto the 1-dimensional Jacobian θ’s. By way of illustration we exhibit a simplestg = 2 non-elliptic potential.

Differential properties of θ-functions described in previous section allows us totake these as a starting point for definition of the functions themselves. It turns outthat such a view leads again to a Liouvillian extension but the latter is accompaniedby introducing a meromorphic elliptic integral and brings up some questions aboutdifferential structures of the multi-dimensional Θ. All this is expounded in Sect. 8.

In Sect. 9 we exhibit some counterexamples fitting no to the canonical FG-theta-theory but being certainly integrable a la Picard–Vessiot with the solvableGalois groups. One of good examples is the famous and fundamental Hermitianequation (containing a parameter) very closely related to the theory of Eq. (1).

Section 10 contains some conclusive remarks.

SPECTRAL/QUADRATURE DUALITY OF FINITE-GAP POTENTIALS 5

2. Background

2.1. Commuting operators. The standard soliton/FG-solutions of integr-able equations are known to be defined through the associated linear PDEs for theauxiliary Ψ-function:

(2) L(U; ∂x)Ψ = λΨ , Ψt = A(U; ∂x)Ψ ,

where L and A are the ordinary differential (scalar or matrix) operators with coef-ficients U being, in general, some functions of variables (x, t): U :=

uk(x, t)

.

The set of function U is usually termed as the potential. This is because thefirst of Eqs. (2) contains no ∂t and thereby may be considered as a spectral prob-

lem defined by the operator expression L. Moreover, the classical (spectral [30])property of the potential U to be finite-gap is determined by the spectrum of thiseigenvalue problem and does not depend on t. On the other hand, the classicalBurchnall–Chaundy–Baker (algebro-geometric) formulation [16, 7] uses one morespectral problem instead of second equation in (2):

(3) L(U; ∂x)Ψ = λΨ , A(U; ∂x)Ψ = µΨ ,

In both the formulations the nontrivial theory appears if equations (2) or (3) arecompatible; there exists a common solution Ψ and the potential is subjected to

certain conditions. These are the well-known commutativities of operators[L, ∂t−

A]= 0 for (2) and

[L, A

]= 0 for (3).

2.1.1. Why one should pass from (2) to (3)? There is a simple explanation

as to this question. Let the operator A be determined by an hierarchy of someintegrable PDEs

(4) Ut = K([U ]),

where, as usual in a formal differential calculus [27] and in the sequel, the symbol [U ]

is used to denote the finite set of derivatives U, Ux, Uxx , . . .. Such hierarchies havebeen well tabulated in the literature [19, 28]. Regard Eqs. (2) from the viewpointof their explicit integration (in some sense of the word). In a straightforwardstatement this problem is impossible to solve because t is a hidden variable inthe first of Eqs. (2). In fact, this variable may be thought of as an additionalspectral one4 and the t-dependence of the potential U is unknown/undetermined.Complexity of the question is not reduced until Eqs. (2) remain partial differentialequations. Indeed, these equations do not have a general solution expressible interms of any known functions. This is because Eqs. (4), integrable as they are, arenot solvable in general. The only way, in order to solve the question, is to considersome particular situations when PDEs (4) admit transformations into some ODEs.Clearly, such a possibility is related to the self-similar reductions of Eqs. (4) andthe most simple case is of course the reduction to the stationary variable z = x−ct.Assuming now the dependence U = U(x− ct), we get, instead of (2),

L(U(z); ∂z)Ψ = λΨ , Ψt = A(U(z); ∂z)Ψ .

The time t disappears in the first of these equations and therefore there existsa solution in form of separability of variables: Ψ = T (t) · ψ(z). Substituting this

4This is so indeed because any additional parameter in coefficients of L(U; ∂x) may beformally viewed as a spectral variable defining a spectral operator pencil.

6 YU. BREZHNEV

ansatz into the last equations, we get immediately a separability parameter5 µ andan exponential dependence T = exp(µt). Equations (2) thus acquire form (3):

L(U; ∂z)ψ = λψ,(A(U; ∂z) + c∂z

)ψ = µψ,

where U = U(z). Compatibility conditions of these equations[L, A + c∂z

]= 0

generate the stationary Lax–Novikov equations F ([U ]) = 0 [28] and, incidentally,different kind reductions may lead to other kind equations. For example, the general

form of Painleve equations and their ‘(L, A)-pairs’ can be obtained by this way [47].2.1.2. Algebraic curve and integrals of Lax–Novikov equations. Let n, m be the

orders of the operators L, A respectively. Assuming that Eqs. (3) are compatible,we conclude, by elimination of the Ψ from (3), that parameters λ and µ are related

by a polynomial dependence W (nµ,

mλ) = 0 and commuting operators L, A them-

selves are also tied by the same dependence: W (A, L) is a zero operator [16, 7].

The identity W (A, L) = 0 implies that its coefficients, being the differential func-tions Ej of U, must be free constants Cj independent of x. These yields a setof (compatible) ODEs Ej([U ]) = Cj providing some integrals of motion Ej for theLax–Novikov equations mentioned above [19]. The further theory requires that thepotential U be the complex analytic function of the complex argument x.

2.2. Theta-function formulae. Solution to Eqs. (3) is an n-valued functionof λ (and an m-valued function of µ) and this multi-valuedness is related to thealgebraic equation W (µ, λ) = 0. This equation, being viewed as an algebraic curveover C, defines a compact Riemann surface R of a finite genus. It is well knownthat multi-dimensional Θ-functions are the universal tool in order to impart a‘single-valued form’ to the analytic apparatus on Riemann surfaces of multi-valuedalgebraic functions [6]. Baker [7], initiated by work [16], was the first to transferthis λ-multi-valuedness of the Ψ into a single-valued function of a point P ∈ R

and to construct the function itself through Riemann’s Θ-functions; the very firstsentence of the work [7] indicates this. Akhiezer [3] arrived at the same objects whenconsidering the problem (1) from completely different—purely spectral—viewpoint.In the 1970s all these discoveries were substantially developed, generalized [29,30, 41, 23, 37, 38, 25], and acquired their current Θ-function form. The recentexcellent survey by Matveev [43] is, perhaps, the best work both on the historyof the question and background material. Most general and modern treatment ofthese results goes back to works by Krichever and reads as follows.

The spectral variable is thought of as the meromorphic function λ = λ(P) inthe sense that the complex number λ in problem (2) is replaced with an abstractcoordinate on R: the point P . The variable x is viewed as a parameter now andthe function Ψ, as function on R, is the function Ψ(P) of an exponential type[3, 7] with essential singularities of some prescribed form [41, 37]. The formularealization of this result is known presently as a concept of the Baker–Akhiezer

(BA) function [8]. The operator L as above and its coefficients (the potential U)is called the finite-gap or algebro-geometric. Of course, we could equally well say

the same about µ = µ(P) for operator A in problem (3).

5An independent treatment of the second eigenvalue of the second operator.

SPECTRAL/QUADRATURE DUALITY OF FINITE-GAP POTENTIALS 7

The structure of solutions is universal [7, 37]. For example, the scalar problem

(5) LΨ :=dn

dxnΨ+ u2(x)

dn−2

dxn−2 Ψ+ · · ·+ un(x)Ψ = λΨ ,

and therefore equation (1), in the class of FG-potentials, has a solution which isgiven, under some normalization, by a general formula:

(6) Ψ(x; λ(P)

)=

Θ(xU +D +U(P)

)

Θ(xU +D)eII(P)x .

Distinctions between different FG-potentials uk(x) are only in the changes of theassociated algebraic curve W (µ, λ) = 0 and the curve itself (its R with a canonicalhomology base (a,b)) determines all the quantities appearing in (6). Namely: Θ(z)is the canonical g-dimensional theta-series

(7) Θ(z) := Θ(z|Π) =∑

N∈Zg

eπi〈ΠN,N〉+2πi〈N,z〉

of g arguments z = (z1, . . . , zg), built by an a-periods Π-matrix of normalizedholomorphic Abelian integrals U(P) =

(U1(P), . . . ,Ug(P)

)on R; symbols like

〈N, z〉 denote Euclidian scalar product 〈N, z〉 := Njzj ; P is a free point on R;II(P) is a normalized Abelian integral of the second kind with only pole of the firstorder at a point P∞ at which λ(P∞) = ∞; the vector U times 2π i is a vector ofb-periods of this integral; D is an arbitrary constant g-vector. All this terminologyis exhaustively expounded in the numerous literature (see, e. g., [6, 8, 18, 24, 25,28, 41]) and the algebraic dependence W (µ, λ) = 0 is referred frequently to asthe spectral curve. As for the problem (1), this curve constitutes a hyperellipticequation of the form

(8) µ2 = (λ − E1) · · · (λ− E2g+1)

and all the FG-potentials are given by the famous Its–Matveev formula [41, 29]

(9) u(x) = −2d2

dx2lnΘ(xU +D) + const.

Example 1. Most simple and popular example is a 1-gap potential. It is uniqueand is determined by the Weierstrass elliptic curve

(10) µ2 = 4(λ− e)(λ− e′)(λ − e′′);

we denote its modulus as Π = ω′/ω, where ω, ω′ are Weierstrassian half-periodsnormalized by the condition ℑΠ > 0 [4]. Since this case is a 1-dimensional one, wereplace P 7→ u and put6 U(P) = u. Therefore, normalizing integral II, we have

II(u) = ζ(2ωu)− 2ηu ⇒

II(u+ 1) = II(u)

II(u+ Π) = II(u) +π

iω⇒ U = − 1

2ω,

where ζ(z) := ζ(z|ω, ω′) and η := η(ω, ω′) are the standard objects accompany-ing the theory of Weierstrassian function ℘(z) := ℘(z|ω, ω′) [4, 55]. Putting forsimplicity D = 0, formulae (6) and (9) become

u(x) = −2d2

dx2lnΘ

(x

2ω

∣∣∣Π)− 2

η

ω, Ψ(x;λ) =

Θ(

x

2ω− u

∣∣∣Π)

Θ(

x

2ω

∣∣∣Π) e

ζ(2ωu)−2ηux

6Jacobian of an elliptic curve is isomorphic to the curve itself.

8 YU. BREZHNEV

and λ = ℘(2ωu) (this is the formula λ = λ(P) above). This potential is a preciseequivalent of the classical Lame form u(x) = 2℘(x − ω − ω′) and 1-dimensionalΘ(z|Π)-series here coincides exactly with the Jacobi function θ3(z|Π) defined by thestandard formula (31).

All the constructions mentioned above—spectral, algebro-geometric, Θ-func-tion, and their varieties—are completely equivalent [38], which is why we shall referto these approaches merely as spectral for short.

3. Integrability of equation (1) by Picard–Vessiot

3.1. The function R. The theory of equation (1) is closely related to thefundamental linear differential equation

(11) Rxxx − 4(u+ λ)Rx − 2uxR = 0

determining function R = R(x;λ). Integrating this equation and denoting an inte-gration constant as µ, we get

(12) µ2 = −1

2RRxx +

1

4Rx

2 + (u+ λ)R2

(see work [13] for a successive derivation of these formulae with use of Lie’s sym-metries approach.) Then the FG-solutions to the Ψ-function for Eq. (1), in generalposition µ 6= 0, are given by the well-known formula

Ψ±(x; λ) =√R(x;λ) exp

x∫ ±µdxR(x;λ)

= exp

x∫Rx(x;λ) ± 2µ

2R(x;λ)dx.(13)

Formulae (11)–(13) and their variations have been repeatedly appeared in the lit-erature [23, 25, 5, 8, 28]. Their precise meaning, however, lies in the fact that theavailability of formula (13) itself does not mean any integrability [13]. This is justan ansatz for Eq. (1) and its solution should be written down in terms of indefiniteintegrals; otherwise all the procedure would reduce to re-notations.

3.2. General formula for the Ψ-function. In the language of commutingBurchnall–Chaundy operators [16] an explicit formula for the Ψ-function (including(13)) results from the sequential elimination of derivatives Ψ(k) from the pair ofdifferential equations (3) down to the formula

(14) Ψx = G([U ];λ, µ)Ψ ⇒ Ψ = exp

x∫G([U ];λ, µ)dx.

This simple recipe of getting the Ψ-function, perhaps, has no received mention inthe modern literature [54]; it is, however, implicitly exploited in monograph [28].Elimination of the last derivative Ψx leads to equation of the curve W (µ, λ) = 0.

The algebraic dependence W (A, L) = 0 serves, in some of works, as the basis for aformal definition of the algebraic integrability [38].

SPECTRAL/QUADRATURE DUALITY OF FINITE-GAP POTENTIALS 9

3.3. Novikov’s equations and differential field. Motivated by the desireto define a differential field over which the integration is performed, we need to knowthe differential structure of function R(x;λ). It is well known that this function isa series in λ with coefficients being differential polynomials in u(x). Computationalformulae for these polynomials have been detailed in the famous work [27, formulae(8)]. Finite order differential conditions on the potential appear if equation (11)has a solution being a polynomial in λ [13]. As in the FG-theory, the equation(11) is also known in differential Galois theory as the second symmetric power ofoperator (1) [50, §4.3.4], [52, p. 671].

Definition 3.1. The potential u(x) is said to be an FG-potential if equation(11) has a solution R(x;λ) being a polynomial in λ. No restrictions on coefficientsof the curve (12) ⇔ (8) have been imposed.

This definition is not a standard one but it is of course equivalent to the spectral[30, Theorem 1], algebraic [27, Ch. 3], or quadrature [21, 13] definitions. The onlyexception is a formal Θ-function (algebro-geometric) setting because it does not usespectra, resolvent R, or quadratures. An important point here is the fact that thefunction domain for the potential is not defined as usual in Galois theory but calcu-lated. This calculation is a problem of nonlinear integration, that is integration ofODEs for the potential u(x). These are the famous Novikov equations [48]. We shallcall this base field the Novikov differential field N ([u]) in u(x)-representation bear-

ing in mind that u(x) satisfies a Novikov equation F (u, ux, uxx , . . . , ux(2g+1)) = 0.

The field N ([u]) consists of rational functions of u(x) over C(λ) and its derivatives.Parameter λ, the g constants ck coming from the integral Gel’fand–Dickey recur-rence [27], and branch points Ek of the curve (12) belong to a subfield of constantsfor N ([u]). Here is an example of Novikov’s equation under g = 2:

(15) (uxxxxx − 10uuxxx − 20uxuxx + 30u2ux) + c1(uxxx − 6uux) + c2ux = 0 .

Integration constant µ, in an FG-class, is fixed to be dependent on the pa-rameter in equation, that is µ = µ(λ), and equation (12) turns into the formula(8). At the same time, as soon as R(x;λ) becomes a polynomial in λ it becomesa differential polynomial R([u];λ) ∈ N ([u]). It should be noted that one suffices tohave only one solution of Eq. (11) belonging to N ([u]) (see Sect. 9.1 further below).

3.4. Picard–Vessiot field, constants, and their hyperelliptic exten-sion. As usual, the Picard–Vessiot extension

N ([u])⟨

Ψ±⟩

:= N ([u])(

Ψ+,Ψ−,Ψx+,Ψx

−)

,

i. e., the splitting field, results from attaching to the field N ([u]) integrals Ψ± ofequation (1) and its derivatives [32, 51, 34, 31, 45]. A simplest kind of extensionscorresponds to solvable cases of the Galois theory and is known as the extension ofLiouville [50, p. 33], [45, 31].

Definition 3.2. An extension N of the differential field N is said to be Liou-villian if there exists a tower of fields N = N0 ⊂ N1 ⊂ · · · ⊂ Nn = N such thatNk+1 = Nk(ψk), where ∂xψk or ∂x lnψk is an algebraic element over Nk.

In other words, Liouvillian extensions are the natural enlargements of the basefield performed by a step-by-step adjunction of solution to the simplest integrableODEs: the 1st order linear ODEs

(16) ψx = Aψ +B

10 YU. BREZHNEV

with coefficients (A,B) being algebraic/rational over previous step field. The stan-dard adjunctions of an algebraic element a, integral

∫pdx, or exponential exp

∫pdx,

where p ∈ Nk, are obtained by putting here (A,B) = (0,ax), (A,B) = (0, p), and(A,B) = (p, 0) respectively.

Solvability of equation by quadratures is directly connected with a structure ofthe group constituting a differential version of the polynomial Galois group.

Definition 3.3. Differential Galois group Gal(N〈Ψ〉

)of a linear ODE de-

fined over N is a set of linear transformations of its solutions Ψ’s that preserve allthe algebraic (over N ) relations among Ψ’s and their derivatives Ψ(n) (differentialautomorphisms group).

The Picard–Vessiot extension must have the same set of constants as N ([u])[32], [34, p. 411], [50]. It is known that for equations of the form (1) the WronskianΨ′

1Ψ2 −Ψ′2Ψ1 is a constant. Taking (13) as one of possible bases for solutions, we

get

(17)Ψx

+Ψ− −Ψ+Ψx− =

√Rx

2 − 2RRxx + 4(u+ λ)R2

= 2µ(λ).

Clearly, it must be a constant of the Picard–Vessiot field whatever the potentialu(x) may be. Neglecting for the moment other constants, we obtain that field ofconstants C(λ) requires adjunction of the constant µ: C(λ) 7→ C(λ, µ). Characterof this extension is determined by the λ-dependence of the function R. It may bepolynomial, rational, or essentially transcendental. The case of rational polynomialR(λ) is possible only if its poles do not depend on x; for we should otherwise havethe λ-dependent poles of the Ψ-function, which is impossible by virtue of structureof Eq. (1). Hence the function R must be an entire function of λ. The Galoisgroup does depend in general on parameters of equation but we are interested inthe following cases:

• When the Lie–Kolchin integrability structure is the same for generic λ?

Therefore two kinds of theories do exist, according as the field C(λ, µ) does not,or does, belong to a finite algebraic or infinite extension of C(λ). The latter fieldsare excessively general because there are huge varieties of entire transcendentalfunctions (without any classification) and they do not produce any restrictions onpotential. On the other hand, finite extensions are of fixed algebraic (necessarily hy-perelliptic) character and calibrated by the only number, namely, by the λ-degree gof the polynomial R([u];λ). In this case infinite Gel’fand–Dickey recurrences termi-nate and equation (12) leads to a finite set of differential restrictions on u(x) in formof differential polynomials Ej([u]) = Cj . This gives in fact yet another indepen-dent motivation (a la Picard–Vessiot) for appearance/availability of a polynomialsolution to Eq. (11). For brevity, we shall adopt however the previous shorter no-tation for the base field N ([u]) without explicit indication of its λ, µ-dependenceN ([u];λ, µ) or dependencies on other field constants N ([u];λ, µ, Cj , . . .), where dotsstand for remaining constants which arise upon complete integration of a Novikovequation.

3.5. Finite-gap Galois groups. Below is a characterization of the Galoisgroup of equation (1) in the class of FG-potentials. Condition on parameter λ of

SPECTRAL/QUADRATURE DUALITY OF FINITE-GAP POTENTIALS 11

being an arbitrary quantity is a fundamental requirement meant throughout thepaper.

Theorem 3.4. The Picard–Vessiot extension N ([u])〈Ψ±〉 is a Liouvillian ex-

tension of the transcendence degree equal to 1. Associated group Gal(N ([u])〈Ψ±〉

),

under generic λ 6= Ej, is connected and isomorphic to the group G =(α 00 α−1

),

where α ∈ C. For other values of λ it is isomorphic to group G =(±1 α0 ±1

).

Proof. For generic λ’s integral (13) does not belong to N ([u]). From (13) itfollows that this extension is Liouvillian. Take the canonical basis of solutions (13).Then the three quantities Ψ−,Ψx

+,Ψx− are expressed through the transcendent

being adjoined Ψ+ as follows:

(18) Ψ− =R

Ψ+, Ψx

+ =Rx + 2µ

2RΨ+ , Ψx

− =Rx − 2µ

2Ψ+(µ 6= 0).

Clearly, in the case of such λ’s that µ = 0 these relations cease to be valid since(Ψ+, Ψ−) become linearly dependent on each other. The relations should be mod-ified and we choose the following basis Ψ±:

(19) (Ψ−)2 = R, Ψx− =

Rx

2RΨ− , Ψx

+Ψ− −Ψ+Ψx− = 1 .

In both of these cases we adjoin one transcendental element (respectively):

Ψ+ = exp

x∫Rx + 2µ

2Rdx or Ψ+ =

√R

x∫dx

R.

In the latter case the radical√R = Ψ− can sometimes be element of N ([u]) as an

example of the Lame equations shows [55, Ch. 23], [40].Let us check invariance of relations (18)–(19) with respect to linear transfor-

mation of the basis (Ψ+

Ψ−

)7→(αΨ+ + βΨ−

γΨ+ + δΨ−

);

this determines the Galois group G =(α βγ δ

)completely. In case (18) we get the

following set of equalities:

αγ = 0 , β δ = 0 , αδ + βγ = 1 , β = 0 .

From this it follows that β = 0, γ = 0, and αδ = 1. The number α can not be anyalgebraic one; for we should otherwise have a finite Galois group and the algebraicΨ±-solutions to give a contradiction with a single-valuedness of the general formula(6). In case (19) we derive that δ2 = 1, γ = 0, αδ = 1, and no conditions on β(except for degenerate cases of curve). Hence in generic case λ 6= Ej the group G

is connected. Its possible forms are thus as follows:

G =(α 00 α−1

)or G =

(δ β0 δ−1

),

where δ = ±1 (compare with cases 3, 4 in Proposition 2.2 of [45]).

To summarize briefly, we conclude that independently of the topological genusof the curve (8), ‘finite-gap’ groups Gal

(N ([u])〈Ψ±〉

)do not depend on parameter

λ and cease to be diagonal and connected only for isolated values of the parameter.

12 YU. BREZHNEV

Corollary 3.5. Equations (1) of the FG-class are factorizable over N ([u]):

∂xx − (u+ λ) =

(∂x +

1

2

Rx

R± µ

R

)(∂x − 1

2

Rx

R∓ µ

R

).

Remark 1. We used nowhere any specific form of the polynomial R. Theo-rem 3.4 is easily restated for arbitrary integrable λ-pencils of the 2nd order withthe only condition that R(x;λ) ∈ N ([u]). Recall that the spectral λ-pencil is a gen-eralization of the canonical spectral equation of the form (5) to more complex (e. g.

polynomial) dependencies of the differential expression L on the external parameter

λ, that is L(U; ∂x, λ) = 0. An example is the well-known spectral λ-pencil of theform

(20) Ψxx − uxu

Ψx −(λ2 − ux

uλ+ uv

)Ψ = 0;

it arises when integrating the integrable nonlinear Schrodinger equation [28, 46, 8].In Sect. 9 we shall consider other examples of the λ-pencils (see also [13, Sect. 5(a)]).

By virtue of structure of the formula (14) Theorem 3.4 has a direct generaliza-tion.

Theorem 3.6. Let N ([U ]) be a Novikov field associated with the pair of Burch-

nall–Chaundy scalar operators (3). Then both of these equations are integrable in

Liouvillian extensions of the same transcendence degree. Under generic λ, µ the

differential Galois groups of these equations are connected and isomorphic to the

diagonal groups G = Diag(α, β, . . . , γ) ⊂ GL(C).

Novikov’s equations are known to be Hamiltonian systems integrable by Liou-ville [19]. However such a way of their integration is not necessary since determi-nation of the potential, i. e., construction of the field N ([U ]), is given by formulaefollowing completely from the ‘linear’ Picard–Vessiot theory. It does not requireHamiltonians.

We finish this section with a digression to one remarkable example. It is aMatveev 1-positon potential given by the seemingly elementary formula [42]

(21) u = −2 lnxxsin(ax+ b)− ax− c

.

Surprisingly, in spite of its complete fitting into the integration scheme above, itis not amenable to integration by means of any classical algorithm in the Picard–Vessiot theory (Kovacic [35], Singer [51]). Indeed, these algorithms are applicableto the finite algebraic extensions of C(x), whereas this u ∈ C(x, ei(ax+b), a, c).

4. Spectral/quadrature duality. An integration procedure

4.1. Drach–Dubrovin equations. The quadrature Drach approach gives avery simple explanation as to why and where the fundamental polynomial

(22) R([u];λ) =(λ− γ1(x)

)· · ·(λ− γg(x)

)

comes from, what its roots γk are, and why these are precisely the quantities thatcomplete the indefinite quadratures7. In what concerns the spectral viewpoint, aremarkable result by Dubrovin [22, 23] is that the quantities γk arise as zeroes ofa Θ-function since they solve the inversion problem of Jacobi [3, 41, 30].

7An elementary explanation to appearance of these objects (supplemented with Russiantranslation of works [20, 21]) can be found in [12].

SPECTRAL/QUADRATURE DUALITY OF FINITE-GAP POTENTIALS 13

Lemma 4.1 (Drach [21], Dubrovin [22]). Functions γk(x) satisfy the system of

ODEs

(23)dγkdx

= 2

√(γk − E1) · · · (γk − E2g+1)∏

j 6=k

(γk − γj), j, k = 1, . . . , g

and potential is determined by the trace formula [29]

(24) u = 2

g∑

k=1

γk(x) −2g+1∑

k=1

Ek .

Proof. One inserts (22) into (12) and takes (8) into account. Collecting theresult in degrees (λ − γk)

n, one requires identity under arbitrary λ. One gets (23)and (24).

Remark 2. We presented such a way of proof because it is applicable to higherorder operators and even spectral λ-pencils. The reason is that the derivation oftrace formulae is not evident when generalizing. Formulae of such a kind ceaseactually to be the ‘trace formulae’ because they have no longer Gel’fand’s treat-ment of the operator trace analogs. They are also not derivable directly from thedefinition of R-polynomial like (22); illustrative counterexamples can be found inwork [11].

4.2. Weierstrass theorem and new representation for the Ψ-function.Main content of this section was briefly announced in [13] and we shall presenthere the extensive proofs, derivations, and precise correlation between spectral andquadrature approaches.

Theorem 4.2. Let u(x) be an FG-potential corresponding to the arbitrary curve

(8). Then solution to equation (1) is given by the quadratures

(25) Ψ±(x;λ) = exp1

2

γ1(x)∫w ± µ

z − λ

dz

w+ · · ·+

γg(x)∫w ± µ

z − λ

dz

w

,

where w2 = (z − E1) · · · (z − E2g+1) and functions γk = γk(x) are determined

through inversion of the set of indefinite integrals

(26)

g∑

k=1

γk∫zg−1 dz

w= 2x+ ag ,

g∑

k=1

γk∫zndz

w= an+1 , n = 0, 1, . . . , g − 2 .

The FG-potential u(x) is determined by formula (24).

Proof. Let us substitute (22) into (13) and change the integration variable xto z. We then obtain the following rules

Rx

Rdx = d ln

∏

k

(λ− γk) 99K1

z − λdz ,

2µ

Rdx =

2µ∏j(λ− γj)

· 12

dγkk

∏

j 6=k

(γk − γj) 99K−µz − λ

dz

w,

wherein 2k = (γk−E1) · · · (γk−E2g+1). Abelian integrals of 3rd kind, as appeared in(25), result from differential equations (23). Furthermore, substitution of expression

14 YU. BREZHNEV

(25) into equation (1) leads, to get an identity, to formula (24); this can serve as yetanother way of derivation of the trace formula. Symmetrizing right hand sides ofequations (23), we rewrite them down in a form that admits an application of theindefinite integration operations, that is (26). This set determines γk as functionsof x.

Corollary 4.3. In u(x)-representation the extension N ([u])〈Ψ±〉 requires the

one quadrature (13). If formula (25) is used then extensions N ([u])〈Ψ±〉 consist in

an adjunction of a symmetric sum of the logarithmic Abelian integrals.

Remark 3 (Definition). Special attention must be given to the fact thatintegrability of the ‘linear Ψ’ is also algebraic (hyperelliptic) as is nonlinear inte-grability of N ([u]). More precisely, in what follows the term ‘algebraic’ will mean

that ultimate answers contain finitely many indefinite integrals of algebraic func-

tions and inversions of the formers.

As for transition from primary λ-dependence to the x-one and vice versa, theduality between spectral and quadrature representations is nontrivial; it is charac-terized by the following statement.

Theorem 4.4. Quadrature and spectral approaches are equivalent in the sense

that the explicit transition between formula (25) and its spectral counterpart (seebelow) is realized through the Weierstrass theorem on permutation of limits and

parameters in a normalized Abelian integral of third kind.

Proof. Sub-exponential expression in (25) is a sum of Abelian integrals, eachwith logarithmic singularities on R at two points: (z, w) = (λ,+µ) and branchplace (z, w) = (∞,∞). According to a Weierstrass theorem, we may exchangesingularities of an elementary logarithmic integral with its limits [6, 18, 41]. Inour case, these are (z, w) = (γ(x), (x)) and a lower limit (z, w) = (α,+β), whereβ2 = (α− E1) · · · (α − E2g+1). More precisely, switching the places

(γ(x), (x)

)

(α, β)

(λ, µ)(∞,∞)

,

we obtain that the difference

γ(x)∫

α

w + µ

z − λ

dz

w−

λ∫

∞

w + (x)

z − γ(x)− w + β

z − α

dz

w= · · ·

is to be everywhere finite quantity, that is certain holomorphic integral:

· · · =γ(x)∫

A1(λ) +A2(λ)z + · · ·+Ag(λ)zg−1 dzw.

Sum of g such quantities must be a holomorphic integral depending symmetricallyon γ’s. Expression (25) is thus converted to its dual object

(27) Ψ± ≃ exp1

2

λ∫w ± 1(x)

z − γ1(x)

dz

w+ · · ·+

λ∫w ± g(x)

z − γg(x)

dz

w+ holomorhic(λ, x)

.

SPECTRAL/QUADRATURE DUALITY OF FINITE-GAP POTENTIALS 15

This is nothing else but the spectral formula by Its & Matveev [30] deserving tobe mentioned more often. We reproduce8 their result as it has been written in [30,p. 351]:

(28) ω(λ) =

λ∫

βn

M(λ)

2√p(λ)

dλ,

αj∫

βj−1

dω(λ) = 0, j = 1, . . . , n,

where M(λ) = λn + a1λn−1 + . . .+ an and

(29) ωk(λ) =

λ∫

∞

(√

P (λ) +√

P (λk(x))

λ − λk(x)−

√

P (λ) +√

P (λk(0))

λ − λk(0)+Mk(λ)

)dλ

2√P (λ)

,

(30) ψ(x, λ) = exp

(ixω(λ) +

n∑

k=1

ωk(λ)

).

Meaning of all the quantities presented in (28)–(30) and transition (25) (30)are obvious from the context. To put it differently, the permutation theorem ofWeierstrass, regarding inverse transition (30) → (25), is a way of doing normaliza-tion of periods of these integrals so that all the parameters in the spectral formulae(28)–(30) can be ‘dumped’ to a common multiplication constant for the Ψ. By thismeans the non-indefinite integrals (27) or (29) with a parametrical dependence ontranscendental functions γk(x) turn into the indefinite ones (25) of an algebraicfunction.

Description of invariant property of equation (1) to be integrable has beencompleted and we conclude the section with the comments about fundamentaldifference between spectral and quadrature modes of getting the formulae.

4.3. On a Riemann surface. At this point not only do analysis on Riemannsurfaces does not come into play, but also the surfaces themselves do not appear.One has just a designation

µ :=√(λ− E1) · · · (λ − E2g+1)

and the theory consists of elementary substitutions (see footnote on p. 3). On theother hand, verification of spectral formulae (28)–(30) is a highly nontrivial tasksince they contain a complete set of transcendental objects of Riemann’s theory ofAbelian integrals and differentiation of an integrand containing γ’s. In this respect,not using the permutation theorem, the ‘spectral integral’ in (27), that is

λ∫

∞

w ± (x)

z − γ(x)

dz

w,

would be very akin to an integral representation of any special function, say, com-plete elliptic Legendre’s integral

K(x) =

1∫

0

dz√(1− z2)(1− x2z2)

.

8We have not found mention of this important result in the literature. See also formula (3)in Akhiezer’s work [3].

16 YU. BREZHNEV

The latter is not expressible by means of any finite Liouvillian extension over C(x)since K(x2) satisfies a 2nd order irreducible 2F1

(12 ,

12 ; 1∣∣x2)-hypergeometric equa-

tion [4].

Remark 4 (history). Both the integrabilities are due to Liouville but chrono-logically, ‘linear integrability’ (1830–40s) was preceded by more famous nonlinearintegrability of Hamiltonian systems (1840–50s). Despite the numerous modernliterature, the fact that these two kinds of integrability are non-casually related tothe one name Liouville was first observed by Morales-Ruiz [45, pp. 51–52].

5. The Θ-functons

By virtue of the fact that extension N ([u])〈Ψ±〉 is transcendental, analyticrepresentations for the previous formulae require introducing new functions. Theseare the Θ-series (7) involved to the theory by Matveev and Its in their famous work[30].

Theorem 5.1. The Θ-function representations (6), (9) are deducible from

quadrature (25). The expressions (6) and (25) are proportional to each other.

Proof of this theorem and consecutive derivation of formula (6) from (25) havebeen detailed in [13, §7]. An important point here is the deductive appearance ofall the aggregates of formula (6) when it is viewed as an axiomatic one: the integralII(P), its periods U , and the Abel map U(P).

From this theorem, we may draw the conclusion that when recognizing the in-tegrability of linear equations the Θ-functions themselves are not necessary. Theyrealize a step to be considered as the next one after emergence of an integral sym-bol

∫in (13) and (26).

5.1. Some differential properties of theta-functions. Theorem 5.1 sug-gests a search for representation of the field N ([u]) by means of Θ-functions. To dothis require some differential properties of Θ-functions and we show further thatthey are available. All the FG-theory tells us that Abelian and BA-functions satisfycertain differential identities. By these identities are meant the fact that Abelianfunctions, as theta-function ratios of linear sections of g-dimensional jacobians ofcurves, have a lot of differential relations between themselves and many of suchrelations have forms of known integrable PDEs [15, 28]. Adding here exponential

functions of the BA-type, we involve into analysis (L, A)-pairs for these PDEs.Moreover, let FG-potential be expressible through the θ-functions of Jacobi. Thennot only do Abelian and BA-functions satisfy certain differential identities but θ-functions themselves also satisfy some ODEs.

Denote by θ[εδ

]the standard θ-series of Jacobi with characteristics (ε, δ) [4]:

(31) θ[εδ

](x|τ) :=

∞∑k

−∞eπi(k+ ε

2)2τ+2πi(k+ ε

2)(z+δ2) ,

i. e., θ[11

]= −θ1, θ

[10

]= θ2, θ

[00

]= θ3, θ

[01

]= θ4. Let ϑ := θ(0|τ) be corresponding

ϑ-constants and θ′1(x|τ) stands for x-derivative of the series −θ[11](x|τ). Period of

the meromorphic elliptic integral is denoted by η = ζ(1|1, τ).

SPECTRAL/QUADRATURE DUALITY OF FINITE-GAP POTENTIALS 17

Theorem 5.2. Jacobian functions θ[εδ

], θ′1 with arbitrary integral characteris-

tics are differentially closed over the field of the (ϑ2, η)-constants and satisfy the

autonomous ODEs

(32)

∂θ[εδ

]

∂x=

θ′1θ[11

] θ[εδ

]− (−1)[

δ2

]

επϑ

[εδ

]2 · θ[ε−10

]θ[

0δ−1

]

θ[11

]

∂θ′1∂x

=θ′12

θ[11

] − π2ϑ[00

]2ϑ[01

]2 · θ[10

]2

θ[11

] − 4η +

π2

12

(ϑ[00

]4 + ϑ[01

]4)· θ[11

],

where[δ2

]signifies an integer part of the number δ/2.

These formulae are consequences of more general differential properties of Ja-cobi’s functions briefly tabulated in [14]. One can see that the similar propertiesare inherent characteristics of the general Θ-functions if the g-dimensional jacobianis isomorphic to a product of elliptic curves. Many examples of such reductions canbe found in [8].

Example 2. Define a Θ-function with characteristics[

αβ

]

as follows:

(33) Θ[

αβ

]

(z|Π) := i〈α,β〉Θ(z +

1

2Πα+

1

2β∣∣∣Π)· eπi〈α,z+ 1

4Πα〉 .

Then in the case g = 2 we have the following identity.

Proposition 5.3. The reduction formula for genus g = 2 under Π12 = 12 :

(34)

Θ[

αεβδ

]

(1

2z− 1

8ατ

w− 1

4α

∣∣∣1

4τ 1

2

1

2κ

)=

= eπ2iα(z+β+ 1

4ατ) ·

θ[0ε

](z|τ)θ[εδ

](w|κ) + i2β+ε · θ[1ε

](z|τ)θ[ ε

δ−1

](w|κ)

.

Differential properties of these Θ, Θ′-functions follow completely from Theorem 5.2.

This example is a rather illustrative one because curves have often symmetriesand if a genus-2 curve has an involutory symmetry differed from the hyperellipticone (λ, µ) 7→ (λ,−µ) then one can show that its Π-matrix is reducible to form (34).Formula (34) is perhaps a simplest case of reduction of the theta-functions to thetwo elliptic tori τ and κ. More complex equivalent of (34) is presented in [8].

Corollary 5.4. Let jacobian of the curve (8) be splittable into a product of

the elliptic curves and the collective symbols θ, ϑ, η stand for arising Jacobi’s

theta-functions and their constants. Then the field C∂(θ;ϑ2, η, . . .) is a differential

extension of N ([u]) (dots indicate other constants of the field).

Example 3. Non-elliptic 2-gap potential (9) with Θ = Θ[0000

]for the reduction

case (34):

(35) u = −2 lnxxθ4(Ux+A|τ)θ2(V x+B|κ) − iθ1(Ux+A|τ)θ1(V x+B|κ)

,

where τ,κ, A,B, V are arbitrary. Since the differential θ-calculus is completelyat hand, we get a particular but nontrivial example of solution to Dubrovin’s ef-fectivization formulae for genus g = 2. Recall that the problem consists [24] indetermination of the ‘winding’ vector U and is described by a system of equationscontaining the undetermined fourth derivatives of the Θ-function [24]. In the ex-ample under consideration the ultimate answer turns out to be quite finite butsomewhat large to display here. Equation for the one sought-for quantity U is analgebraic equation of degree 9 (exercise: derive it).

18 YU. BREZHNEV

5.2. Linearly exponential divisor and Θ-representation of N ([u]). Letus use notation of formula (6) and plug into (9) an inessential exponential multiplier:

(36) u(x) = −2d2

dx2lnΘ(xU +D) ehx + const.

Introduction of this term is motivated by the fact that solutions for the Ψ-function(6) are expressed not only through the Θ-functions but involve an exponentialfactor. We shall call the quantity Θ(xU+D) ehx, with h, U , and D being constantswith respect to ∂, the linearly exponential divisor or Lx-divisor. Let us form a fieldC(Lx). The following proposition gives a weaker property than Corollary 5.4, butit is valid for arbitrary genera.

Proposition 5.5. The field C(Lx

)is ∂-differential and finitely generated.

Proof. Expression (36) satisfies a Novikov equation which has finite order

2g + 1. Hence the derivatives dn

dxnΘ(xU + D) ehx of order n > (2g + 1) + 2 areexpressed rationally through Lx and its lower derivatives.

The field Θ∂ = C∂

(Θ(xU +D)

)may be considered as a field generated by one

linear divisor (h = 0). It is obviously that N ([u]) ⊂ Θ∂. Of course, Θ∂ containsnow not only Abelian functions but this extension is well defined since Θ-seriesand b-periods U are computed once u(x) has been given. For this reason, we canredefine equation (1) as one given over Θ∂ and thereby we let

(37) N (Θ) := C∂

(Θ(xU +D)

).

Although N ([u]) $ N (Θ), the field N (Θ) is said to be a Θ-representation ofNovikov’s fields. The constant λ, for the moment, may be disregarded.

It should be emphasized that in the FG-integration there appears not merelyan abstract multi-dimensional Θ-series but its specification Lx; the 1-dimensionallinear section of jacobians. Moreover, the most general Θ-function is an objectdefined up to an exponential multiplier (see (33)) so that we may think of it, andtherefore of the divisor Lx, as a continual generalization of Θ-function with discrete

characteristics. Thus, Liouvillian solutions N ([u])〈Ψ±〉 are expressed through theΘ with a linear dependence of its arguments upon x and the ‘linearly exponential’multiplier9 eII(P)x first explicitly appeared in Akhiezer’s work [3] and subsequentlywas axiomatized by Krichever [37].

Remark 5. In what follows we shall exhibit that the continually parametricobject Lx may be introduced in its own right, in a particular case, through the quad-

rature integrable ODEs. Interestingly, the different kind sections of theta-functionarguments can lead to other important equations. One remarkable property of sucha kind appears even in the g = 1 case. Let us consider the function θ(x|τ) as afunction on a simplest (i. e., straight line) section of the 2-dimensional variety ja-cobian ⊗ moduli space. Without loss of generality we may impart to this functionthe form θ1(Aτ + B|τ). Then this object generates the general Hitchin class ofsolutions to the sixth Painleve equation in a form exactly as does the finite-gapformula (9), that is logarithmic derivative of a ratio of entire functions [14].

9This ‘linear exponent’ is a result of the contemporary theory; Baker [7] did not specify anexponential Θ-structure of solutions but just cited to pp. 275, 289 of his [6].

SPECTRAL/QUADRATURE DUALITY OF FINITE-GAP POTENTIALS 19

6. Integration as a linearly exponential Θ-extension

In this section we give a formulation of the Θ-function scheme as integra-tion a la Picard–Vessiot. Let us consider the above Picard–Vessiot extension [50]N ([u]) ⊂ N ([u])〈Ψ±〉 in the representation u(x). This transcendence is Liouvillianand dependence of the Ψ on parameter λ is also transcendental contrary to the‘rationality’ of the field C(λ, µ). Meanwhile, based on Theorems 3.4 and 5.1, we seethat the field has the following structure:

N ([u])〈Ψ±〉 = C∂

(Θ(

xU +D +U(P))

Θ(xU +D)eII(P)x

).

It immediately follows that if we pass to the Θ-representation (37) then integrationprocedure can be reduced to one operation. Namely, a field, over which an equa-tion has been defined, is supplemented with an element of the same form as onegenerating the field itself:

(38) N ([u]) ⊂ N (Θ) ⊂ C∂

(

Θ(xU +D) ehx, Θ(xU +D +U(P)) eII(P)x)

.

It is significant in this viewpoint that problem of the ‘linear integration’ dropsout along with the problem of building the base field N ([u]) being treated as aproblem of the nonlinear integration. In the Θ-representation the potential is de-termined only by means of operations in the field N (Θ); formula (36).

Theorem 6.1. All the embeddings

N ([u]) ⊂ N (Θ) ⊂ N ([u])〈Ψ±〉 ⊂ C∂

(

Θ(xU +D), Θ(xU +D +U(P)

)eII(P)x

)

are the Liouvillian extensions.

Proof. The fact that embedding N ([u]) ⊂ N (Θ) is Liouvillian follows directlyform formula (9). The statement concerning the second embedding is a consequenceof (13) because R([u];λ) ∈ N (Θ). Rewriting formula (6) in the form

Θ(xU +D +U(P)

)eII(P)x = Ψ+ ·Θ(xU +D),

we deduce that property for the last embedding to be Liouvillian results from theproportionality of (6) and expression (13) for Ψ+ (Theorems 4.2 and 5.1).

Theorem 6.2. For generic λ integration of equation (1) in the Θ-representa-

tion is equivalent to a multiplication of an element Ξ(x) generating the field N (Θ)by an adjoined linearly exponential divisor:

Proof. Consider Ξ(x) = Θ(xU +D)−1. It is clear that C∂(Ξ) = N (Θ). Then

(39) Ψ±(x;λ(P))= C± · Ξ(x) ·Θ

(xU +D ±U(P)

)e±II(P)x

because all the holomorphic/meromorphic integrals on hyperelliptic curves changesign under permutation of sheets µ 7→ −µ; we may write ±U(P), ±II(P) in (39).

This theorem has an important treatment:

• When passing to the Θ-representation the equation (1) is integrated as if it

were an equation with constant coefficients. Integration procedure is thus

trivialized under a proper choice of ‘domain of rationality’. The inverse

transform method for the soliton class is a particular case of this general

construction.

20 YU. BREZHNEV

Indeed, the simplest FG-case corresponds to the 0-gap one with N (Θ) = C(λ) andwe need only one linear exponent; the construction (39) acquires the form

(40) Ψ±(x;λ) = C± · Ξ(x) · e±a(λ)x , Ξ(x) = e0·x ∈ N (Θ),

wherein Ξ(x) has the ‘same form’ as the adjoint element ea(λ)x. Adjoining all theexponents associated with an N -soliton solution (and their varieties like positons(21) or rational solitons), we obtain the general 0-gap case.

Remark 6. The structure of solution (39) in form of simple multiplicationof elements generating N (Θ) and its extension N ([u])〈Ψ±〉 is not quite typicalfor equations integrable by attaching the linear exponents [35] or, especially, forequations with a solvable Galois group [32]. This property owes its origin to theavailability of λ in equation (Theorem 3.4).

Transition between u- and Θ-representations is transcendental with respect toλ-dependence and other constants of the field. These constants are the Π-matricesof curves, U -periods, and vector D. We may therefore trivialize the scheme aboveif we proceed further and redefine equation (1) over (37) as one given over theλ-pencil (field) of the Lx(P)-divisors:

Lx(P) := Θ(xU +D +U(P)

)eII(P)x .

Although field C∂

(Lx(P)

)is generated by the infinite number of elements, equation

(1) itself constitutes an infinite λ-pencil of equations. Strictly speaking, both spec-tral and quadrature approaches require to look upon Eq. (1) as being a differentially-

algebraic one: differential in x and algebraic (polynomial) in λ. Furthermore, byvirtue of Proposition 5.5, the arbitrary Lx(P)-divisor generates some solution ofNovikov’s equation. It may be fixed by choice of one element of the pencil Lx(P):

(41) N (Θ) → C∂

(

Θ(xU +D0 +U(P0)

)eII(P0)x

)

.

Having extended the field (41), that is C∂

(Lx(P0)

), to the field C∂

(Lx(P)

), its

Galois group becomes trivial since the integral of equation (1) is given now in formof a ratio of two field elements.

7. Integrability and differential closedness

7.1. Differential closure in terms of θ-functions. Attaching the divisorLx(P) as transcendental element with a parameter P tells us that it should be

introduced to the theory as the base function, along with the available Θ’s withoutparameter P . Owing to Theorem 6.2 this would arrive us at a closed differentialapparatus (differential closedness) accompanying spectral equation and potential.Presently, the general Θ-formula realization of this viewpoint is an open problem,which is why we illustrate it by cases when g-dimensional Θ-function reduces to acombination of Jacobian ones.

At first glance, from Theorem 5.2, it would seem that supplement of the basis(32) with Lx-divisor of the type θ1(x − u) ehx requires also adjunction of all thefunctions θ′1, θ2,3,4(x − u). That no such complication takes place will be apparentfrom the following statement.

SPECTRAL/QUADRATURE DUALITY OF FINITE-GAP POTENTIALS 21

Theorem 7.1. For Weierstrassian curve (10) with modulus τ = ω′/ω one de-

fines an elliptic Lx(P)-divisor Λ by the formula

Λ(x; u|τ) := θ1(x− u|τ) exp(θ′1(u|τ )

θ1(u|τ )x+ hx

), u /∈ Zτ + Z,

where h is an extra parameter. Then the six functions Λ, θ′1, and θk (k = 1, 2, 3, 4)satisfy the closed autonomous system of ODEs over the field C

(η, ϑ2, θ(u)

):

∂θk∂x

=θ′1θ1θk − πϑ2k ·

θnθmθ1

, n =8k − 28

3k − 10, m =

10k − 28

3k − 8

∂θ′1∂x

=θ′12

θ1− π2ϑ23ϑ

24 ·θ22θ1

− 4η +

π2

12

(ϑ43 + ϑ44

)· θ1

1

Λ

∂Λ

∂x=θ′1θ1

+πϑ22θ1(u)

· θ31(u) · θ2θ3θ4 + θ2(u)θ3(u)θ4(u) · θ31

θ1 ·(θ22(u) · θ21 − θ21(u) · θ22

) + h

.(42)

Motivation for the theorem lies in the fact that Abelian integrals, on the onehand, are representable in terms of theta-functions and, on the other hand, aredifferentially closed: the base integrals of 1st, 2nd, and 3rd kind form a differentialbasis. Indeed, derivatives of integrals are functions and any function is expressedthrough the two base ones (℘, ℘′) (generators of an elliptic functions field) whichare in turn integrals of exact meromorphic differentials of 2nd kind. However theta-function representation for a 3rd kind integral is still lacking in system (32).

Proof. Meromorphic functions are at hand since they are formed by θ-quo-tients. Derivatives of meromorphic functions are again meromorphic ones but dif-ferentiation of θk(x) generates θ′1(x). The quotient θ′1/θ1 is proportional to theWeierstrass ζ-function which in turn represents a meromorphic (i. e., 2nd kind) el-liptic integral [4]; it is alone as genus g = 1. Without loss of generality we may setthat the missing integral of 3rd kind has two logarithmic singularities. The place ofone of them may be fixed at x = 0 and the second place can be taken as a parame-ter. Call it u. Residues of a corresponding differential are opposite in sign and theycan be moved to a common multiplication constant. Such an integral III is unique(even for arbitrary g) up to a holomorphic one(s) since all the other integrals areexpressed through III = III(x; u). Its derivatives are again meromorphic functionsand the process closes. It will suffice to add an exponent of III and integral III itselfis given by the known formula

(43) III(x; u) :=1

2

z∫w + w0

z − z0

dz

w= ln

σ(2x− 2u)

σ(2x)e2ζ(2u)x

,

where z = ℘(2x), w = ℘′(2x) and z0 = ℘(2u), w0 = ℘′(2u). Weierstrassian pa-rameters (ω, ω′), presented in (43), are replaced by the one quantity τ thanks tohomogeneity relation ω2℘(ωx|ω, ω′) = ℘(x|1, τ) =: ℘(x|τ). Holomorphic integral isabsent in system (32) but is present in a basis of Abelian integrals. Missing elementΛ0 can be formally adjoined by setting Λ0(x|τ) = x and supplementing system (42)with equation dΛ0

dx= 1. Adding to (43) the holomorphic integral hΛ0 and convert-

ing the right hand side of (43) to the θ-functions, one obtains that adjunction ofexp(III) is equivalent to adjunction of the object Λ(x; u|τ). Differentiating (43) andconverting it to the θ’s, one arrives, after some simplification, at the last equalityin Eqs. (42).

22 YU. BREZHNEV

If u ∈ Zτ + Z then the integral III turns into a logarithm of meromorphicfunction: ln(℘(x) − e). There is nothing to adjoin.

Corollary 7.2. Let potential be a finite-gap one and g-dimensional jacobian

split into a product of the elliptic curves. Then differentiation of g-dimensional

functions Θ(xU +D

)reduces to a set of the ‘1-dimensional’ equations (42). The

functions θk, θ′1 and Λ, taken possibly with different moduli τ , form the differentially

closed basis over which every Novikov’s equation of order 6 2g+1 and problem (1)are integrated.

In the framework of this corollary integrability of Novikov’s equations is amanifestation of differential closedness of the first two equations in (42). The thirdequation in (42) ‘integrates’ equations with a parameter and their consequences(see examples in [13]). Here, ‘integrates’ means that all these equations are nothingmore than combinations of system (42) and its derivatives. Let us consider someexamples.

Example 4. The two gap Lame potential

Ψxx =24℘(2x|τ) + λ

Ψ .

This is a classical example presented in many places [8, 28]. Corresponding solutionexpressed in terms of our objects reads as follows:

(44) Ψ± =d

dx

Λ(x;±u|τ)θ1(x|τ)

, h =±2µ

3λ2 − 122g2(τ), ℘(2u|τ) = λ3 + 123g3(τ)

36λ2 − 123g2(τ),

µ2 =(λ2 − 48g2(τ)

)(λ3 − 36g2(τ)λ + 432g3(τ)

) (⇔ (8)

),

where g2(τ), g3(τ) are the standard modular τ -representations for Weierstrassianparameters a, b entering into the curve w2 = 4z3 − az − b [4]. Novikov’s equation(15) is satisfied under (c1, c2) = (0,−672g2(τ)) and R-polynomial has the form

R([u];λ) = λ2 − 1

2uλ+

1

4u2 − 36g2(τ),

where u = 24℘(2x|τ).

7.2. Non-elliptic example. We consider here a potential being no ellipticfunction but expressible through the ‘elliptic’ θ-functions.

Example 5. The Ψ-function for non-elliptic potential (35). It is a nonlinearsuperposition of the one-gap Ψ-functions. If we denote for brevity z = Ux+A andw = V x+B we then derive that

(45) Ψ(x;λ) =Λ(z+1

2 τ ;U1

∣∣τ)Λ(w+ 1

2 ;U2

∣∣κ)− Λ(z;U1|τ)Λ(w;U2|κ)

θ1(z+1

2 τ∣∣τ)θ1(w+ 1

2

∣∣κ)− θ1(z|τ)θ1(w|κ)

eII(λ)x

,

where holomorphic integrals U1, U2 are the certain linear combinations Uk = Ckjuj

of the elliptic holomorphic ones u1, u2 because the curve (8) corresponding to theΘ-function (34) has the form

(46) µ2 = λ(λ− 1)(λ− a)(λ− b)(λ − ab) =: P5(λ)

SPECTRAL/QUADRATURE DUALITY OF FINITE-GAP POTENTIALS 23

and is realized as covers of two tori defined by moduli τ and κ:

(47)

℘(2u1|τ) + ϑ42(τ) + ϑ33(τ) = 3(1− a)(1− b)λ

(λ − a)(λ − b)ϑ43(τ),

℘(2u2|κ) + ϑ42(κ) + ϑ33(κ) = 3(1− a)(1− b)λ

(λ − a)(λ − b)ϑ43(κ).

These are the classical formulae by Jacobi [8] presented in terms of Weierstrass’ ℘and the pair of branch points a, b and moduli τ,κ are commonly written downfor one another [8]. All the information concerning this curve can be found in [8]and we omit details of some calculations. We need to compute the integral II(λ).

Abelian integrals for the curve (46) are expressed through θ, θ′1 and thereforederivation of the meromorphic integral II(λ) is a routine calculation. An explanationis that the reduction of holomorphic integrals to elliptic ones entails the reduction ofthe meromorphic Abelian integrals to the meromorphic elliptic ones. The latter areexpressed through Weierstrassian ζ-function, i. e., θ′/θ, and meromorphic ellipticfunctions. First translate formulae for cover (47) into the language of θ-functions:

(48)ϑ22(τ)

ϑ23(τ)

θ24(u1|τ)θ21(u1|τ)

=(1− a)(1− b)λ

(λ− a)(λ− b),

ϑ22(τ)

ϑ23(τ)

θ24(u1|τ)θ21(u1|τ)

=ϑ22(κ)ϑ23(κ)

θ24(u2|κ)θ21(u2|κ)

.

This is a complete set of equations determining the holomorphic integrals u1, u2 asfunctions of λ. The point λ = ∞ corresponds to the values u1 = 1

2 τ and u2 = 12 κ.

Drop out for the moment indication of modulus τ in the following transformationof a meromorphic elliptic integral:

∫sds√

4s3 − g2s− g3=

∣∣∣∣∣s =π2

12

(ϑ43 + ϑ44 − 3ϑ23ϑ

24

θ23(u1)

θ24(u1)

)∣∣∣∣∣

= −1

2

θ′1(u1− 1

2 τ)

θ1(u1− 1

2 τ) − 2ηu1 = · · ·

On the other hand, the 1st formula in (48) supplemented with use of the standardquadratic ϑ, θ-identities brings this integral into the following expression:

· · · = const

∫s · λ−

√a√b√

P5(λ)dλ ≃

∫ ((λ− a)(λ − b)

(1 − a)(1 − b)λ+ϑ42 + ϑ433ϑ43

)λ−

√a√b√

P5(λ)dλ,

that is meromorphic integral on the curve (46) with a pole at λ = ∞ and a surplusone at λ = 0. Doing the same for the second torus (u2) with modulus κ, weobtain one more meromorphic integral with the same infinities. Forming their linearcombination, we can construct the integral with a single singularity at infinite point.After some algebraic simplifications the sought-for result becomes:

(49)

II(λ) = a · θ′1

(u1−1

2 τ∣∣τ)

θ1(u1− 1

2 τ∣∣τ) + b · θ

′1

(u2−1

2 κ∣∣κ)

θ1(u2−1

2 κ∣∣κ) + c ·u1 + d ·u2

= a · θ′1(u1|τ)θ1(u1|τ)

+ b · θ′1(u2|κ)θ1(u2|κ)

+(λ+ p)µ

λ(λ− a)(λ− b)+ q ·u1 + r ·u2 ,

where constants a, b, c, d, p, q, r depend only on parameters of the potential (35),i. e., on τ , κ, A, B, V , and are independent of λ. Again, after careful co-ordinationof all the moduli and normalizing constants the direct check of (1), (35), (45), and(49) becomes a good exercise in a differential θ-calculus. We mention in passing

24 YU. BREZHNEV

that this example cannot be elaborated in the framework of the standard ellipticsoliton theory [8].

7.3. A new treatment of the spectral parameter. In proof of Theo-rem 7.1 parameter u was the only ‘external’ parameter independent of ‘internal’parameters of the curve (moduli). On the other hand, the only parameter beingexternal to the field N ([u]) and equation (1) is λ. It plays an isolated role. Apartfrom the fact that it is merely present in equation, it may be treated as an objectarising from the two independent ‘mechanisms’: 1) adjunction of a transcendentalelement and, on the other hand, 2) differential closedness of all the Abelian inte-grals. As evidenced by the foregoing and formula (25), these are the same things:

• The logarithmic singularity in a canonical integral of third kind is arbitrary

and independent of moduli. The property of the theory to be integrable is,

by construction, independent of it. It may therefore always be thought of

as a (free) spectral variable.

The converse is also true. Differentiation of the 3rd kind integrals yields otherintegrals and functions. In other words, informally speaking, one may say that

• Closed class of ODEs integrable through Θ(xU +D) is in fact integrable

in terms of Θ, Θ′, Lx and ‘owes’ to contain an external (except for

moduli) parameter P.

Indeed, there is one fundamental logarithmic integral III for each algebraic curveand it has a single parameter P ⇔ λ. In the elliptic case this III has form (43) and forarbitrary genera it is expressed through the g-dimensional Θ’s [6]. The one fold log-arithmic ∂x-derivative of (25), i. e., derivatives of III(γ′s;P), yields the rational func-tions of (z, w)(γ′s) and therefore only meromorphic objects remain since integralsthemselves disappear. As a rough guide we have here ∂x exp(III) = IIIx · exp(III).This linear in exp(III) equation10 is treated as a spectral one. Coefficients of thisequation (more precisely its λ-independent pieces) can be thought of as potential.The quantity P should always be distinguished as an external one because, other-wise, the following chain

external λ ⇔ ∂x-closedness ⇔ ‘Ψ-linearity’

is destroyed altogether. Moreover, the x-dependence Θ(xU +D) is not bound tobe a linear one and spectral equations must not necessarily be of the form (3).Counterexamples in Sect. 9 illustrate these points.

8. Definition of θ through Liouvillian extension

Insomuch as the object Λ(x; u|τ) is in fact a theta-function with a parameter,we can use the differential equations described above as the basis for a definition ofthe theta’s themselves and, in particular, consider character of their integrability.

Proposition 8.1. System (32) has the two algebraic (rational) integrals

ϑ22 θ24 − ϑ24θ

22 = A1ϑ

23 θ

21 , ϑ22θ

23 − ϑ23θ

22 = A2ϑ

24θ

21

generalizing the famous Jacobi θ-identities when A1 6= 1 6= A2.

10An important remark is in order. We might not say the same as applied to the ‘purespectral’ object (27) since non-indefinite integral remains. Again, Weierstrass’ theorem does thejob. The ‘one-fold ∂x’, which is equivalent here to the ‘∂x-differential closedness’, explains why

the spectral equations are always linear.

SPECTRAL/QUADRATURE DUALITY OF FINITE-GAP POTENTIALS 25

Proof. The straightforward calculation shows that ∂xA1 = ∂xA2 = 0.

In a nutshell, the differential genesis of the object Λ is as follows. Function θ′1is determined differentially through θ1. Therefore two functions with two arbitraryconstants solve the system (32). One of constants serves a homogeneity θ 7→ Cθ of(32). Another one u is non-algebraic and is related to an autonomy of equations(32). Hence the two transcendental functions θ1(x − u|τ) and θ2(x − u|τ) remain.These functions are represented, up to a shift and holomorphic integral, by the oneobject Λ(x; u|τ).

Theorem 8.2. Differential equations (42) are algebraically integrable.

Proof. By a direct computation one can show that any solution θk = θ of thesystem (42) satisfies the same 5th order ODE

(50)

(1

Fx

(Fx

2

F

)

x

)

x

+8Fx = 0 , F = (ln θ)xx − 2κ, −κ := 2η+1

6π2 (ϑ43 +ϑ44).

Therefore, not taking into account u, there is only one essential parameter in equa-tions (42), i. e., parameter κ. From this it follows that

F = Ξ(x;a, b, c) :

F∫dz√

z(z − a)(z − b)= 2ix+ c

and integration is thus completed if, according to definition in Remark 3, we adjointhe inversion operation Ξ:

(51) θ = exp

x∫ x∫

Ξ(y;a, b, c)dydx · eκx2+dx+e ,

where a, b, c,d, e are the integration constants. Of course, the inversion function Ξhere bears no relation to ratios of the θ-series. Integration of equations for functionsθ′1 and Λ is obvious.

Remark 7. In a separate work we shall show that the algebraic integrabilityabove can be supplemented with a Hamiltonian formulation X = Ω∇H(X) to thesystem (42) and its Lagrangian description.

It is noteworthy, that variable θ satisfies an equation of fifth order, not third, asit would be expected from the well-known ℘-equation of Weierstrass [4]. Anotherpoint that should be mentioned here is the fact that algebraic integrability of equa-tions (42) leads not merely to the θ-function itself but to the elliptic Lx-divisor andeven its non-canonical extension by the quadratic exponent. (Notice that constantκ depends on modulus but constant d is free.) Further, the two-fold integration ofthe transcendental inversion operation in (51) can be reduced to integration of analgebraic function—our base operation.

Corollary 8.3. The θ-function can be defined through a meromorphic elliptic

integral.

To prove this it will suffice to make the following substitution in formula (51):

x∫Ξ(y;a, b, c)dy =

Ξ(x;a,b,c)∫zdz√

z(z − a)(z − b).

26 YU. BREZHNEV

By this we obtain somewhat nonstandard way of introduction of the θ-functions.To all appearances, Tikhomandritskiı [53] was the first to point out a way of defini-tion of the θ through a meromorphic integral11 but his note [53] went unnoticed inthe literature. He poses a question about the natural going from elliptic integralsto the theta-functions and presents the mode of transition between these transcen-dents by introducing the integral of the 2nd kind elliptic integral. Indeed, rewritingformula (51) in the following form

(52) θ(x) = exp

x∫ Ξ(x)∫

zdz√

z(z − a)(z − b)

dx · eκx2+dx+e ,

we observe that such a way of introduction of a θ-function is in effect the result

of Liouvillian extension of a meromorphic integral, i. e., adjoining an exponent of

integral of such an integral. By this means we may adopt this point as a differ-

ential definition of the θ a la Liouville and, subsequently, construct all the otherobjects of the theory: meromorphic (algebraic) functions are the θ-ratios, Abelianintegrals of 2nd kind are expressed through just introduced meromorphic integral(or, which is the same, the θ′), and the 3rd kind integrals are the logarithmic ratiosof the θ’s with free parameters (the Λ-objects). Holomorphic integrals are of coursethe independent objects; they are not defined/determined through any other ones.An important role of a meromorphic integral was already observed by Clebsch &Gordan in preface to their book [18, p. VI] on the base of Jacobi’s formula

Θ(u)

Θ(0)= exp

u∫

0

Z(u)du,

where Z(u) is a Jacobi zeta-function notation in the theory of elliptic functions [4].The above differential properties of the Θ’s splittable to θ’s raise the question as

to whether the general multi-dimensional Θ-functions admit the similar ‘differentialkind’ definition. In particular, whether exists the purely Liouvillian definition ofan x-section Θ(xU + D) like formula (52)12 or, if any, the closed set of partial

DEs defining the complete set of the general Θ, Θ′(z)-functions as ones of the garguments z? Some relations between Θ’s and meromorphic Abelian integrals canbe found in lectures by Weierstrass (though no really this point has been mentionedin the modern literature) but the question about closed and differentially Liouvillian

definition (if it exists) of a 4g-set of the g-dimensional Θ-functions and associatedderivatives Θ′ remains an important open problem.

11He does not mention the quadratic extension and differential closedness of the set θk, θ′1,

however.12Roughly speaking, one needs an extensive strengthening of Theorem 6.1; whence it follows

that

−2 lnxxΘ(xU + D) = lnxxΨ + (lnxΨ)2 + const

and, since the Ψ is an exponent of the 3rd kind Abelian integral (formula (25)), that isΨ = expIII(γ’s), the object Θ(xU + D) itself is computed as a ‘Liouvillian extension of Abelianintegrals’:

Θ(xU + D) = exp−1

2

III(γ’s) +

∫∫ [III(γ’s)

]2xdxdx

eax2+bx+c

which is a reminiscence of formula (51).

SPECTRAL/QUADRATURE DUALITY OF FINITE-GAP POTENTIALS 27

9. Non-finite-gap integrable counterexamples

Definition of integrability domain is not a subject of the Θ-function techniques.Therefore we may generate integrable equations by any way differed from the classi-cal FG-structure defined by formula (6) and Theorem 5.1. For example, equations

coming no from operators L(U; ∂x) by taking the canonical spectral equation

L(U; ∂x)Ψ = λΨ form in general the operator λ-pencils L(U; ∂x, λ)Ψ = 0, say,(20). Their solution structure is not known a priori13. Moreover, we can evenconstruct an equation fitting no in the FG-scheme but having the same formalΘ-function form of solution.

Example 6. Omitting in notation the elliptic modulus τ , elucidate the said aboveby the following modification of the 2-gap Lame equation:

Ψxx =24℘(2x) + 8℘(2x− u) + 16℘(u)

Ψ .

It has a solution of the formal 2-gap Baker–Akhiezer form (44):

(53) Ψ(x; u) =d

dx

Λ(x; u)

θ1(x), h = 4ζ(u)− 2ζ(2u)

(exercise: check this solution). By Theorems 7.1 and 8.2 this example is alge-braically integrable over C∂

(℘(2x), ℘(2x− u)

)(and over C∂(θ1,Λ), of course) with

solvable Galois group but it has little in common with commutative Burchnall–Chaundy operators, BA-function, or spectral lacunae. Formula (53) shows thatthis Ψ-function has no even a pole at point 2x = u where potential does. This poledepends on a spectral parameter u lying on an elliptic curve.

Nevertheless this example should not be considered as ‘too artificial’ becausethere exist the Θ-function integrable models having algebraic curves with x-de-pendent branch-points. Ernst’s equations in general relativity [36] provide a niceexample along these lines.

9.1. Hermite’s operator pencil. Consider now, in the framework of Picard–Vessiot theory, Hermitian equation (11) itself. It is not a Burchnall–Chaundy op-erator but the operator λ-pencil. By virtue of formulae (11)–(13), we define thispencil over field N ([u]) and repeat arguments about λ-dependence of R.

One of solutions to this pencil is not exponential but a purely Abelian mero-morphic function (22), i. e., differential polynomial

R1 = R([u];λ) =: P ∋ N ([u]).

Since base of solutions to Eq. (11) is Ψ+2 , Ψ+Ψ−, Ψ−

2 and Ψ+2 /∈ N ([u]), we may

put the second solution as a square of the BA-function R2 = Ψ+2 , where Ψ+ is an

adjoint transcendent

Ψ+ := exp

x∫Px + 2µ

2Pdx.

The third solution is R3 = Ψ−2 and we obtain that

(54) R1 = P, R2 = Ψ+2 , R3 =

P2

Ψ+2 .

13It is well known, however, that Eq. (20) is related to a matrix canonical eigenvalue problem.

28 YU. BREZHNEV

Hence rationality domain is the same as in Theorem 3.4, that is N ([u])〈Ψ+〉, andthis extension is a Liouvillian one of the transcendence degree 1 (we consider onlythe generic case λ 6= Ej). We therefore can obtain the following result.

Theorem 9.1. Under the generic λ 6= Ej the Galois group of Hermite’s equa-

tion (11) defined over N ([u]) is connected and isomorphic to the diagonal group

G = Diag(1, α, α−1). Equations (11) is factorizable over field N ([u]).

Proof. As in proof of Theorem 3.4 we perform a linear transformation of thebasis R1, R2, R3 and check invariance of the base differential relations betweenR’s. Let us take relations of the zero and 1st order in derivatives:

R1 = P, R2R3 = P2 , (R2)x =

Px + 2µ

PR2 , (R3)x =

Px − 2µ

PR3 ,

which result from properties (54). Using condition µ 6= 0, one easily derives thatadmissible transformations are

R1 7→ 1 ·R1 , R2 7→ α · R2 , R3 7→ δ · R3

and R2R3 7→ 1 · R2R3. Hence δ = α−1. Solutions R1, R2, R3 are in generalnot algebraic functions, hence α is a free nonzero complex number and we do notneed further to analyze the remaining relations of second order in derivatives ofR’s (they will be automatically satisfied). This yields a connectivity of the groupand the matrix14 Diag(1, α, α−1). Factorizations of Eq. (11) are deducible by use ofLiouville’s scheme since we know solutions to equation. For example

∂xxx − 4(u+ λ)∂x − 2ux =(∂x +

Px + 2µ

P

)∂x

(∂x − Px + 2µ

P

)

=(∂x +

Px + 2µ

P

)(∂x − 2µ

P

)(∂x − Px

P

).

These factorizations are not unique because equation has order 3.

9.2. Inversion of non-holomorphic integrals. As a last counterexamplewe consider an equation that leads, on the one hand, to a nonstandard case ofthe inversion problem, and, on the other hand, to a nonlinear x-evolution in atheta-function argument. As we shall see, there is no essential difference between(quadrature) integrability schemes of this example and those of pure FG-potentials.This example was already pointed out as non-standard in [54].

Example 7. Let us consider the following spectral problem

(55) Ψxx =λ

v2Ψ ,

where v = v(x). As long as we have deal with invariant integration of (55) (Sect. 3),the theory has just non-essential modifications and we restrict ourselves to writingdown all of its attributes in a form of references source for the simplest but nontrivialcase g = 1. We put

(56) R(x;λ) = vλ− φ(x), φ(x) := 2a(x− b)(x− c),

14All this can also be seen from the fact that Galois group belongs to SL3(C) and transcen-dence degree of the extension is unity.

SPECTRAL/QUADRATURE DUALITY OF FINITE-GAP POTENTIALS 29

where a, b, c ∈ C. Novikov’s equation is the equation v3vxxx −4φvx+4φxv = 0 andits integrals are as follows:

(57) µ2 = λ3 + 3E2([v])λ2 + E1([v])λ+ a2(b− c)2 ,

3E2 = −1

2vvxx +

1

4vx2 − 2

φ

v, E1 =

1

2φvxx − 1

2φxvx +

φ2

v2+ 2av .

’Trace formula’ follows from a direct analogy of (22), i. e., we set R = (λ − γ)v,but inversion problem becomes an inversion of the logarithmic integral rather thanJacobi problem (26). Indeed, manipulations with integrals E1, E2 show that

(58)

γ∫1

z − E2

dz√4z3 − g2z − g3

=1

2a(b− c)lnx− b

x− c+D,

where g2 = 12E22 − 4E1 and g3 = 4E1E2 − 8E2

3 − a2(b− c)2.From (58) it follows that the θ-function description undergoes changes since

x-evolution on jacobian is essentially non-linear. We pass from parameter E2 to by the rule ℘(2) = E2 and represent the logarithmic integral (58) in terms ofθ-functions of the holomorphic one r: γ = ℘(2r). We arrive at a transcendentalequation determining function r = r(x):

lnθ1(r − )

θ1(r + )+ 2

θ′1()

θ1()r =

℘′(2)

a(b− c)lnx− b

x− c+D.

As a result we obtain that ultimate solution to the Ψ-function is far from obvious:

Ψxx =λ℘2(2r)

a2(x − b)2(x− c)2Ψ , Ψ±(x;λ) =

√(x− b)(x − c)√

℘(2r)· Λ(r;±u)

θ1(r),

λ = ℘(2u)− ℘(2), h = 2ηu2 .

A direct check of this solution is a good exercise in theta-calculus. Moreover,nonlinearity of this x-evolution is two-fold. Logarithm (58) contains a fraction-linear function but the principal nonlinearity comes from a transcendental nonlin-earity of r(x). It never becomes linear even though we replace the general case in(56), that is φ(x) = 2a(x− b)(x− c), with the particular one φ(x) = const.

10. Concluding remarks

The properties of θ-functions outlined above differ in a crucial respect fromclassical special functions since the latter ones are defined by ODEs not integrablein quadratures over elementary or algebraic functions. For example Bessel’s func-tions or the Painleve transcendents. Therefore when generalizing rational theory(solitons), not only do algebraic integrability takes place for Novikov’s equationsbut it also takes place for linear spectral equations and even θ-functions. In allthese cases the integration procedure has been closed at a single and common step:adjunction of the inversion operation Ξ. The elementary theory does not get bywithout inversion as well:∫

rational functions 99K ln 99K inversion 99K exponent 99K solitons.

It should be also emphasized that it makes no difference whether 1st kind in-tegrals (Jacobi problem) or 2nd, 3rd kind ones have been inverted (see, e. g., (58)).The only thing is needed for the (Liouvillian) algebraic integrability: inversion ofindefinite integrals of any algebraic functions. Roughly speaking, the inversion

30 YU. BREZHNEV

procedure appearing in ‘theta-methods’ has also the Liouvillian characterizationbecause, according to important Eq. (16), adjunction of any kind Abelian integralsabove is allowed. Nontrivial examples on inversion of meromorphic integrals can befound in monograph [28]; they are associated with the Camassa–Holm hierarchyand have also the theta-function description. In the same place quite extensivebibliography is presented. In other words, in regard to invariant integrability, the

choice of the Θ-series or the inversion operation Ξ is just a question of nomencla-

ture. Introducing Θ is equivalent to removing γ’s from formulae like (23)–(25) andconversely. As for analytic representation of solutions, the Θ-series is of course thefundamental object.

References

[1] Acosta-Humanez, P. B. Galoisian Approach to Supersymmetric Quantum Mechanics. The

integrability analysis of the Schrodinger equation by means of differential Galois theory. Ver-lag Dr. Muller, 2010.

[2] Acosta-Humanez, P. B., Morales-Ruiz, J.-J. & Weil, J.-A. Galoisian Approach to In-

tegrability of the Schrodinger Equation. Rep. Math. Phys. 67(3) (2011), 305–374.[3] Akhiezer, N. I. A continuous analogue to orthogonal polynomials on a system of intervals.

Soviet Math. Dokl. 2 (1961), 1409–1412.[4] Akhiezer, N. I. Elements of the Theory of Elliptic Functions. Amer. Math. Soc., Providence,

RI, 1990.[5] Al’ber, S. I. Исследование уравнений Кортевега–де Фриза методом рекуррентных

соотношений. (Investigation of equations of Korteweg–de Vries type by method of recurrence

relations). Journ. London Math. Soc. (2) XIX(3) (1979), 467–480 (in Russian).[6] Baker, H. F. Abelian Functions. Abel’s theorem and the allied theory of theta functions.

Cambridge Univ. Press, Cambridge, 1897.[7] Baker, H. F. Note on the foregoing paper, “Commutative ordinary differential operators,”

by J. L. Burchnall and J. W. Chaundy. Proc. Royal Soc. London A 118 (1928), 584–593.[8] Belokolos, E. D., Bobenko, A. I., Enol’skii, V. Z., Its, A. R. & Matveev, V. B. Al-

gebro-Geometric Approach to Nonlinear Integrable Equations. Springer Series in NonlinearDynamics, Springer–Verlag, Berlin–Heidelberg–New York, 1994.

[9] Беркович, Л. M. Факторизация и преобразования дифференциальных уравнений.РХД, Москва, 2002; Berkovich, L. M. Factorization and transformations of differential

equations. RCD, Moscow, 2002 (in Russian).[10] Braverman, A., Etingof, P. & Gaitsgory, D. Quantum integrable systems and differential

Galois theory. Transformation Groups 2(1) (1997), 31–56.[11] Brezhnev, Yu. V. Dubrovin equations for finite-gap potentials. Russ. Math. Surveys 57(2)

(2002), 415–417.[12] Брежнев, Ю. В. Исторические замечания к теории конечнозонного интегрирования.

http://arXiv.org./nlin.SI/0504051 (in Russian); Brezhnev, Yu. V. Historical remarks to

the theory of finite-gap integration.

[13] Brezhnev, Yu. V. What does integrability of finite-gap or soliton potentials mean? Phil.Trans. Royal Soc. Ser. A 366(1867) (2008), 923–945.

[14] Brezhnev, Yu. V. A τ -function solution of the sixth Painleve transcendent. Theor. Math.Phys. 160(3) (2009), 1615–1632.

[15] Buchstaber, V. M., Enolskii, V. Z. & Leykin, D. V. Kleinian functions, hyperelliptic Ja-

cobians and applications. Review in Mathematics and Mathematical Physics 10(2). Eds.S. Novikov & I. Krichever. Gordon and Breach, London, (1997), pp. 1–125.

[16] Burchnall, J. L. & Chaundy, T. W. Commutative Ordinary Differential Operators. Proc.Royal Soc. London. Ser. A 118(780) (1928), 557–583.

[17] Chudnovsky, D. V. & Chudnovsky, G. V. Appendix I: Travaux de J. Drach (1919). Clas-

sical and Quantum Models and Ariphmetic Problems. Lecture Notes in Pure and AppliedMathematics. Eds. D. V. Chudnovsky & G. V. Chudnovsky. Marcel Dekker, New York, 1984,pp. 445–453.

SPECTRAL/QUADRATURE DUALITY OF FINITE-GAP POTENTIALS 31

[18] Clebsch, A. & Gordan, P. Theorie der Abelschen Functionen. Druck und Verlag vonB. G. Teubner, Leipzig, 1866.

[19] Dickey, L. A. Solitons Equations and Hamiltonian Systems. 2nd edn. Advance Series inMath. Physics 26. World Scientific, Singapore, 2003.

[20] Drach, J. Determination des cas de reduction de l’equation differentielled2y

dx2= [ϕ(x)+h]y.

Compt. Rend. Acad. Sci. 168 (1919), 47–50.

[21] Drach, J. Sur l’integration par quadratures de l’equationd2y

dx2= [ϕ(x) + h]y. Compt. Rend.

Acad. Sci. 168 (1919), 337–340.[22] Dubrovin, B. A. Inverse problem for periodic finite-zoned potentials in the theory of scat-

tering. Funct. Anal. Appl. 9 (1975), 61–62.[23] Dubrovin, B. A. Periodic problems for the Korteweg–de Vries equation in the class of finite

band potentials. Funct. Anal. Appl. 9 (1975), 215–223.[24] Dubrovin, B. A. Theta functions and non-linear equations. Russ. Math. Surveys 36(2)

(1981), 11–92.[25] Dubrovin, B. A., Matveev, V. B. & Novikov, S. P. Non-linear equations of the Korteweg–

de Vries type, finite-zone linear operators and Abelian varieties. Russ. Math. Surveys 31(1)(1976), 59–146.

[26] Ehlers, F. & Knorrer, H. An algebro-geometric interpretation of the Backlund transfor-

mation for the Korteweg–de Vries equation. Comment. Math. Helv. 57(1) (1982), 1–10.

[27] Gel’fand, I. M. & Dikii, L. A. Asymptotic behaviour of the resolvent of Sturm–Liouville

equations and the algebra of the Korteweg–de Vries equations. Russ. Math. Surveys 30(5)(1975), 77–113.

[28] Gesztesy, F. & Holden, H. Soliton Equations and Their Algebro-Geometric Solutions.

(1 + 1)-Dimensional Continuous Models. Cambridge studies in advanced mathematics 79,Cambridge University Press, Cambridge, 2003.

[29] Its, A. R. & Matveev, V. B. Hill’s operator with finitely many gaps. Funct. Anal. Appl. 9(1975), 65–66.

[30] Its, A. R. & Matveev, V. B. Schrodinger operators with finite-gap spectrum and N-soliton

solutions of the Korteweg–de Vries equation. Theor. Math. Phys. 23(1) (1975), 343–355.[31] Kaplansky, I. An introduction to differential algebra. Hermann, Paris, 1957.[32] Kolchin, E. R. Algebraic matrix groups and the Picard–Vessiot theory of homogeneous linear

ordinary differential equations. Annals of Math. 49 (1948), 1–42.[33] Kolchin, E. R. Abelian Extensions of Differential Fields. Amer. J. Math. 82(4) (1960), 779–

790.[34] Kolchin, E. R. Differential Algebra and Algebraic Groups. Academic Press, New York, 1973.[35] Kovacic, J. J. An algorithm for solving second order linear homogeneous differential equa-

tions. Journ. Symbolic Comput. 2(1) (1986), 3–43.[36] Korotkin, D. A. & Matveev, V. B. Theta Function Solutions of the Schlesinger System

and the Ernst Equation. Funct. Anal. Appl. 34(4) (2000), 252–264.[37] Krichever, I. M. Methods of algebraic geometry in the theory of non-linear equations. Russ.

Math. Surveys 32(6) (1977), 185–213.[38] Krichever, I. M. Commutative rings of ordinary linear differential operators. Funct. Anal.

Appl. 12 (1978), 175–185.[39] Lax, P. D. Periodic Solutions of the KDV Equation. Comm. Pure and Appl. Math. XXVIII

(1975), 141–188.[40] Maier, R. S. Lame polynomials, hyperelliptic reductions and Lame band structure. Phil.

Trans. Royal Soc. Ser. A 366(1867) (2008), 1115–1153.[41] Matveev, V. B. Abelian functions and solitons. Preprint n. 373, Wroc law University, 1976,

pp. 1–98.[42] Matveev, V. B. Positons: slowly decreasing analogues of solitons. Theor. Math. Phys.

131(1) (2002), 483–497.[43] Matveev, V. B. 30 years of finite-gap integration theory. Phil. Trans. Royal Soc. Ser. A

366(1867) (2008), 837–875.[44] McKean, H. P. & van Moerbeke, P. The spectrum of Hill’s equation. Inventiones Mathe-

maticae 30(3) (1975), 217–274.[45] Morales-Ruiz, J.-J. Differential Galois theory and non-integrability of Hamiltonian sys-

tems. Progress in Mathematics 179, Birkhauser Verlag, Basel, 1999.

32 YU. BREZHNEV

[46] Newell, A. C. Solitons in Mathematics and Physics. CBMS-NSF Regional Conference Seriesin Applied Mathematics 48, SIAM, Philadelphia, PA, 1985.

[47] Nijhoff, F., Hone, A. & Joshi, N. On a Schwarzian PDE associated with the KdV hierar-

chy. Phys. Lett. A 267 (2000), 147–156.[48] Novikov, S. P. The periodic problem for the Korteweg–de Vries equation I. Funct. Anal.

Appl. 8(3) (1974), 236–246.[49] Previato, E. Multivariable Burchnall–Chaundy theory. Phil. Trans. Royal Soc. Ser. A

366(1867) (2008), 1155–1177.[50] van der Put, M. & Singer, M. F. Galois Theory of Linear Differential Equations. Springer–

Verlag, Berlin, 2003.[51] Singer, M. F. Liouvillian solutions of nth order homogeneous linear differential equations.

Amer. Journ. Math. 103(4) (1981), 661–682.[52] Singer, M. F. Liouvillian solutions of linear differential equations with Liouvillian coeffi-

cients. Journ. Symbolic Comput. 11(3) (1991), 251–273.[53] Тихомандрицкiй, М. А. Замѣтка о введенiи Θ-функцiй въ теорiю эллиптическихъ

функцiй. Сообщенiя и протоколы заседанiй математическаго общества при Император-

скомъ Харьковскомъ Университетѣ (1883), Сер. I, 47–67. Tikhomandritskiı, M. A. A

note on introduction of Θ-functions to the theory of elliptic functions. Communications andproceedings of Math. Soc. of the Khar’kov Imperial Univ. (1883), Ser. I, 47–67 (in Russian).

Brief version of this note: Tichomandritzky, M. Ueber das Umkehrproblem der elliptischen

Integrale. Math. Ann. XXII(3) (1883), 450–454.[54] Ustinov, N. V. & Brezhnev, Yu. V. On the Ψ-function for finite-gap potentials. Russ.

Math. Surveys 57(1) (2002), 165–167.[55] Whittaker, E. T. & Watson, G. N. A Course of Modern Analysis: An Introduction to the

General Theory of Infinite Processes and of Analytic Functions, with an Account of the

Principal Transcendental Functions. Cambridge University Press, Cambridge, 1996.

E-mail address: brezhnev@mail.ru